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Basis FunctionIn functional analysis and its applications, a function space can be viewed as a vector space of infinite dimension whose basis vectors are functions not vectors. This means that each function in the function space can be represented as a linear combination of the basis functions. Let's illustrate the concept of basis using a simple example. You can create any (two-dimensional) vector, (x,y) by adding multiples of the vectors (1,0) and (0,1): -
\begin{bmatrix}x\\y\end{bmatrix} = x \begin{bmatrix}1\\0\end{bmatrix} + y\begin{bmatrix}0\\1\end{bmatrix}. In this example, we say that the vector (x,y) is in the space spanned by the vectors (1,0) and (0,1). The most convenient basis vectors are perpendicular or orthogonal to each other, which is true of (1,0) and (0,1). Two vectors are orthogonal if their scalar product is zero, which means they are at right angles. Likewise, two functions are orthogonal if their inner product is zero. Sine and cosine are orthogonal functions because - .
A function f(x) is square integrable if and only if - .
Any square integrable function (for example a musical recording) can be represented by a sum of sines and cosines of various amplitudes and frequencies. This is termed the function's continuous Fourier transform. In this example, the sines and cosines are the basis functions. Note that while the two-dimensional plane is spanned by only two basis vectors, a function space is spanned by an infinite number of basis functions, because the function space is infinite-dimensional.
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